Of course, the proof (logic)is there as soon as you clearly DEFINE something. But not every layman can follow all "if-thens" from fundamental definition to complex phenomena. Thus, he needs an explanation in more understandable for him terms. And math is just such set of terms and logical consequences of definitions. We call these consequences equations. If we speak about equations resulting in important relationship between natural phenomena, we call those equations "natural laws" (say, energy conservation law, which actually is mathematical identity to symmetry of time).

So, natural phenomena go by logical way which we discover and wrote in symbols and call it "math".